Convex Hull Asymptotic Shape Evolution
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
University of Illinois, Urbana, IL 61801, USA; (MA: Institute for Information Transmission
Problems, Moscow, Russia). Email: {mda,ymb,lavalle}@uiuc.edu
Summary. The asymptotic properties of Rapidly exploring Random Tree (RRT) growth in
large spaces is studied both in simulation and analysis. The main phenomenon is that the convex hull of the RRT reliably evolves into an equilateral triangle when grown in a symmetric
planar region (a disk). To characterize this and related phenomena from flocking and swarming, a family of dynamical systems based on incremental evolution in the space of shapes is
introduced. Basins of attraction over the shape space explain why the number of hull vertices
tends to reduce and the shape stabilizes to a regular polygon.
1 Introduction
Rapidly exploring Random Trees (RRTs) [9] have become increasing popular as a
way to explore high-dimensional spaces for problems in robotics, motion planning,
virtual prototyping, computational biology, and other fields. The experimental successes of RRTs have stimulated interest in their theoretical properties. In [10], it was
established that the vertex distribution converges in probability to the sampling distribution. It was also noted that there is a “Voronoi bias” in the tree growth because
the probability that a vertex is selected is proportional to the volume of its Voronoi
region. This causes aggressive exploration in the beginning, and gradual refinement
until the region is uniformly covered. The RRT was generalized so that uniform random samples are replaced by any dense sequence to obtain deterministic convergence
guarantees that drive dispersion (radius of the largest empty ball) to zero [11]. The
lengths of RRT paths and their lack of optimality was studied in two recent works.
Karaman and Frazzoli prove that RRTs are not asymptotically optimal and propose
RRT*, which is a variant that converges to optimal path lengths [5]. Nechushtan,
Raveh and Halperin developed an automaton-based approach to analyzing cases that
lead to poor path quality in RRTs [12]. More broadly, there has also been interest in
characterizing convergence rates for other sampling-based planning algorithms that
use random sampling [3, 4, 8].
In this paper, we address one of the long-standing questions regarding RRT behavior. Figure 1(a) shows how an RRT appears in a square region after 390 iterations.
2
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
(a)
(b)
Fig. 1. (a) An RRT after 390 iterations. (b) An RRT grown from the center of a “large” disc,
shown with its convex hull.
Fig. 2. The Voronoi diagram of RRT vertices contains interior Voronoi regions, which are
bounded by other Voronoi regions, and exterior Voronoi regions, which extend to the boundary
of the space.
Convex Hull Asymptotic Shape Evolution
3
What happens when the size of the search space is increased? In the limiting case in
which an RRT sequence is generated in a large disc in the plane, simulations reliably
produce three straight tree branches, roughly 120 degrees apart. See Figure 1(b). In
terms of Voronoi bias, the exterior Voronoi regions completely dominate because the
probability that random samples fall into interior Voronoi regions shrinks to zero;
see Figure 1. Thus, the RRT spends virtually all of its time expanding, rather than
refining in areas it has already explored.
It is remarkable that the convex hull of the RRT throughout the process is approximately an equilateral triangle with random initial orientation. This has been consistently observed through numerous simulations, and it known for over a decade with
no rigorous analysis. How can this behavior be precisely characterized? Why does
it occur? This motivates our introduction of convex hull asymptotic shape evolution,
CHASE , which is a family of dynamical systems that includes the RRT phenomenon
just described.
In addition to understanding RRTs, we believe that the study of CHASE dynamics
encompasses a broader class of problems. Another, perhaps comparable in importance, motivation for our research comes from the the general desire to understand
the self-organization in large, loosely interacting collectives of agents, whether natural, or artificial. While the literature on flocking, swarming and general dynamics
of large populations is immense, it is worth noticing that it is predominantly concentrated on the locally interacting agents. Recently, a new trend emerged, dealing
with agents interacting over arbitrary distances: as an example, [14] deals with the
scale-invariant rendezvous protocols in swarms, while naturalists provide us with the
evidence that scale-invariant (topological, in their parlance) interactions in animal
world [1].
2 Dynamics of Shapes
2.1 Rapidly exploring trees
The Rapidly exploring Random Tree (RRT) is an incremental algorithm that fills a
bounded, convex region X ⊂ Rd in the following way. Let T (V, E) denote a rooted
tree embedded in X so that V ⊂ X and every e ∈ E is a line segment (with e ⊆ X).
An infinite sequence (T0 ⊂ T1 ⊂ T2 ⊂ . . .) of trees is constructed as follows. For
T0 (V0 , E0 ), let E0 = 0/ and V0 = {xroot } for any chosen xroot ∈ X, designated as the
root. Each Ti is then constructed from Ti−1 . Let ε > 0 be a fixed step size. Select a
point xrand uniformly at random in X and let xnear denote the nearest point in Ti−1
(in the union of all vertices and edges). If xnear ∈ Vi−1 and kxnear − xrand k ≤ ε,
then Ti is formed by Vi := Vi−1 ∪ {xrand } and Ei := Ei−1 ∪ {enew } in which enew is
the segment that connects xnear and xrand . If kxnear − xrand k > ε, then an edge of
length ε is formed instead, in the direction of xrand , with xnew as the leaf vertex. If
xnear 6∈ Vi−1 , then it must appear in the interior of some edge e ∈ E. In this case,
e is split so that both of its endpoints connect to xnear . Recall Figure 1(a), which
shows a sample RRT for X = [0, 1]2 . The RRT described here uses the entire swath
4
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
for selection, rather than vertices alone, as described in [9]. For a discussion of how
these two variants are related, see [11]; the asymptotic phenomena are independent
of this distinction.
2.2 Isotropic process
Consider the situation when the starting point xroot is at the origin, and the region to
be explored (and the sampling density) is rotationally invariant. In the limit of small
ε, the RRT growth is governed by the direction towards the (far away) xrand , which
can be assumed to be uniformly distributed on the unit circle or directions. In this
case, the dynamics of the convex hulls of the successive RRTs can be described as
follows.
Let Pn denote the convex hull of the points x1 , . . . , xn ∈ R2 . The random polygon
Pn+1 depends on x1 , . . . , xn only through Pn via the following iterative process:
Algorithm 1 Dynamics of the convex hulls of early RRTs in the limit of small step
sizes.
t ← 0, P0 ← 0
loop
l ← rand(S1 )
v ← argmaxhl, xi, x ∈ Pt
Pt+1 ← conv(Pt ∪ {v + εl})
t ← t +1
end loop
This algorithm defines an increasing family of (random) convex polygons. One
might view it as a (growing) Markov process on the space of convex polygons in R2
starting with the origin P0 .
It is easy to see that as the Markov chain evolves, the number of vertices in Pn
can change. Some vertices can disappear, being swept over by the convex hull of the
newly added point. It is easy to see that this new point does not eliminate the vertex
vl where the functional h·, li attains its maximum if and only if the vector −l does
not belong to the cone Tvnl Pn , the tangent cone to Pn at vl . This, in turn, can happen
only if the angle at vl is acute (less than straight).
Enigma of the symmetry breaking
While the distribution of this growth process is manifestly rotationally invariant, the
simulations show that after long enough time the polygons Pn are not getting more
and more round. Au contraire, it was observed that the polygons Pn become close
(in Hausdorff metric1 ) to some large equilateral triangles. (It should be noted that the
1
Recall that the Hausdorff metric between two subsets of a metric space is defined as
dH (A, B) = maxa∈A minb∈B d(a, b) + maxb∈B mina∈A d(a, b).
Convex Hull Asymptotic Shape Evolution
5
polygons are actually triangles only for a fraction of the time; typically some highly
degenerate - with an interior angle close to π - vertices are present.)
While we still do not understand completely the mechanisms of the symmetry
breaking and of the formation of the asymptotical shapes, there are several asymptotic results and heuristic models that shed enough light on the process to at least
make plausible explanations of the observed dynamics. This note is dealing with
these results.
2.3 Roots of difficulties
The biggest problem emerging when one attempts to analyze the Markov process
{Pn }n=1,2,... stems from the fact that there is no natural parameterization of the space
of the polygons: an attempt to account for all convex planar polygons at once leads
immediately to an infinite dimensional (and notoriously complicated) space of convex support functions with C0 norm-induced topology. Indeed, starting with any
polygon, the Markov process will have a transition, with positive probability, changing the collection of vertices forming the convex polygon Pn .
Fig. 3. Typical behavior of the number of vertices of Pt , starting with a regular 50-sided
polygon.
50
40
30
20
10
0
0
500
1000
While the experimental evidence suggests that the polygons Pn become close to
equilateral triangles in Hausdorff metric, there is little hope to understand the process
of the vertices of Pn , as the very number of the vertices is changing constantly, see
Fig. 2.3
2.4 Outline of the results
For this reason, we adopt a circumvential route here. In lieu of addressing the complicated dynamics defined by the algorithm ?? throughout, we concentrate here on
6
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
the time intervals where the number of the vertices of Pt remains constant. During each such stretch, the evolution of the convex polygons P can be described as a
relatively straightforward (time-homogeneous) Markov process on the space of the
k-sided polygons.
If the number of vertices of polygons Pt were to stabilize, then the size of these
polygons would be growing linearly in time (more precisely, the circumference
would)2 . This suggests to consider a Markov process on the space of shapes of ksided polygons (we discuss the state space of this new process, that is the space of
shapes of k-sided polygons in section 2.5).
This new Markov process on the space of shapes cannot be time-homogeneous
anymore, but would be well approximated by the steps of size 1/t. Markov processes
on manifolds with diminishing step sizes are quite familiar in the theory of stochastic
perturbations, where we represent the Markov process as a shift along a vector field
(obtained by the averaging) perturbed by small random noise.
We will describe this averaged dynamical system (which we will refer to as
CHASE , for Convex Hull Averaged Stochastic Evolution) in section ??. Interestingly,
it can be interpreted (up to a time reparameterization) as the gradient field for the
circumference of the polygonal chain: CHASE dynamics tries to make the polygonal
chain longer the fastest way!
It is well known that for small steps, the stochastically perturbed processes are
tracing closely their deterministic, averaged counterparts for long times. More to
the point, the classical results on the stochastic approximation imply that perturbed
dynamical system remains trapped with probability one near an exponentially stable
equilibrium of the averaged dynamics, if the quadratic deviations of the stochastic
perturbation scale like 1/t (or any other scale such that the quadratic variation is
summable as t → ∞), see [7, 13].
Thus the first natural step to try to explain the lack of polygonal shapes other
than equilateral triangles in the long runs of Pt would be to analyze the averaged
dynamics of our Markov processes on the polygonal chains, locating their equilibria
and studying their stability.
As it turned out, for k-sided polygonal chains, the only nondegenerate fixpoints
of the averaged dynamics are the regular polygons3 . Moreover, we prove that these
equilibria are unstable for k ≥ 3.
While the Markov chain Pt is well-approximated by CHASE for k-sided polygonal chains with obtuse internal angles, it is not the case in general (because Pt has
non-zero probability of changing the number of sides). We present some results that
heuristically lend support to the natural conjecture that the regular triangles are stable
shapes under any generalized averaged dynamics.
We conclude the paper with a short description of limit of CHASE dynamics for
the obtuse k-sided polygons, as k → ∞.
2
3
We conjecture that this is true in general, but do not have a proof yet.
The averaged CHASE dynamics can be extended to non-convex polygons; the regular polygons therefore can be non-convex ones as well.
Convex Hull Asymptotic Shape Evolution
7
2.5 Spaces of shapes
We begin with the formal description of the space of shapes.
construction
A configuration of n points is a collection of distinguishable (labelled) points in R2 ,
not all coincident. Traditionally, the shape of a configuration is understood as its
class under the equivalence relation on the configurations defined by the Euclidean
motions and the change of scale. More specifically, two configurations {x1 , . . . , xn }
and {x10 , . . . , xn0 } are said to have the same shape if for some rotation U ∈ SO(2), the
vector v ∈ R2 and λ > 0, x0j = λUx j +v. Factorings by the scale and the displacement
admit sections: one can assume that the center of gravity of the configuration is at
the origin, while it moment of inertia is 1 (or that ∑k |xk |2 = 1) - this is where the
condition that not all points coincide is important. For a natural interpretation of the
space shape in terms of symplectic reduction, see [6].
Complex projective spaces
Perhaps most explicit way to think about the spaces of shapes of planar k-configurations
is the following: viewing R2 as C, we can interpret the k-configuration with the center of mass at the origin as a point in Ck−1 − {0}. Factoring by rotations and rescaling is equivalent to factoring by C∗ , the multiplicative group of complex numbers,
whence the space of shapes is isomorphic (as a Riemannian manifold) to CPk−2 , the
(k − 2)-dimensional complex projective space.
Fig. 4. Sphere of shapes of triangles. The equator consists of degenerate triangles (with three
points collinear). The poles correspond to equilateral triangles.
8
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
The simplest nontrivial case k = 3 is very instructive: CP1 is isometric to the
Riemannian sphere S2 . The poles of the sphere can be identified with the equilateral
triangles (North or South depending on the orientation defined by the cyclic order
of x1 , x2 , x3 ); the equator corresponds to the collinear triples, and the parallels to the
level sets of the (algebraic) area, considered as the function of the configurations
rescaled to the moment of inertia 1 (see Figure 4).
Tame configurations and the k-gon dynamics
The space of planar k-configurations contains an open subset consisting of tame ones,
which we define here as the configurations for which any two consecutive sides form
the positively oriented frame (in particular, all points of the configuration are distinct), and the angle between these vectors is acute:
(xk − xk−1 ) × (xk+1 − xk ) > 0, (xk − xk−1 ) · (xk+1 − xk ) > 0.
The space of tame configurations is invariant with respect to rotations and dilatations,
and its image in the shape space confk = CPk−2 is an open contractible set (say,
for k = 3 it is the upper hemisphere of S2 ). We will denote the space of tame k+
configurations as conf+
k , and the corresponding subset of the shape space as confk .
+
The following observation is obvious: If Pt ∈ confk then for the step size ε
4
small enough Pt+1 ∈ conf+
k . Equivalently, the Markov dynamics on large enough
configuration preserves tameness, for a while.
As one of our goals is to disprove that the Markov process Pt can ever converge
to a k > 4-sided polygon, we can just define a new Markov process which would
couple with Pt on tame configurations.
(k)
We define the tamed Markov process Pt is defined by the algorithm
Algorithm 2 Tamed dynamics.
(k)
t ← 0, P0 = (v1 , . . . , vk ) ∈ confk
loop
l ← rand(S1 )
(k)
i ← argmaxhl, xi, vi ∈ Pt
vi ← vi + εl
t ← t +1
end loop
3 Tamed Dynamics
The tamed Markov process possesses several features that simplify its treatment considerably, compared to the original Markov process. First and foremost, the tamed
4
And tame enough!
Convex Hull Asymptotic Shape Evolution
9
dynamics stays, by definition, in the space of k-sided polygonal chains, isomorphic
to R2k .
Further, the fact that the number of the “growth points” is bounded implies, immediately, that the size of the polygonal chain grows as Θ (t).
Denote by P(P) the perimeter of (not necessarily convex) polygon P.
Lemma 1 The expected perimeter of P(k) grows linearly: for some c,C > 0
P P(P(k) t ) < cn ≤ exp(−Ct).
(Remark that a linear upper bound on the perimeter of P(k) t and of Pt is quite immediate.)
3.1 Continuous dynamics
The stochastic process P(k) can be viewed as a deterministic flow on the space VPk
of k-sided polygons subject to small noise. Indeed, by the Lemma 1, the size of the
polygons P(k) grows linearly, whence the relative size of the steps decreases.
Given a polygonal k-chain P, consider the expected increment
∆ P := E(P(k) t+1 |P(k) t = P) − P
(we use here the linear structure on VPk ). The following is immediate:
Lemma 2 The function ∆ commutes with the rotations and is homogeneous of degree 0.
The linear growth of the sizes of polygons P(k) together with the homogeneity
of ∆ indicate that asymptotically, the expected increments are small compared to the
size of P(k) . In such a situation, an approximation of the discrete stochastic dynamics
by a continuous deterministic one is a natural step. This motivates the following
Definition 1 The vector field
Ṗ = ∆ P
on VPk is called the CHASE dynamics.
In coordinates, the CHASE dynamics is given as follows.
Corollary 1 Assume that P consists of the vertices (x1 , . . . , xk ), xi ∈ R2 in cyclic order, and (xi1 , xi2 , . . . , xic ) are the vertices of P on its convex hull. Then
x˙il =
xi − xii−1
xil − xii+1
+ l
,
|xil − xii+1 | |xil − xii−1 |
and ẋi = 0 if xi is not on the convex hull of P.
(1)
10
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
3.2 Theorem: discrete converges to continuous
The intuitive proximity of the P(k) dynamics and the trajectories of CHASE flow can
be made rigorous, using the standard results. Specifically, we have
Theorem 1 Let P ∈ VPk , and P(k) t , 0 ≤ t ≤ Cλ be the (random) trajectory of the
tamed Markov dynamics with the initial configuration λ P. Similarly, let P(t), 0 ≤
t ≤ λ C be the trajectory of the CHASE dynamics starting with the same initial point
λ P. Then for any C > 0, the trajectory P(k) t /λ converges to P(t)/λ in probability in
C0 norm, as λ → ∞.
The proof requires some relatively extensive setup and will not be presented here.
However, it is conceptually very transparent: the deviations of the stochastic dynamics from the continuous one has linearly growing quadratic variations, and, by the
martingale large deviation results (using, for example, Azuma’s inequality), one can
prove the proximity of the trajectories for the times of order λ with probability exponentially close to 1.
In other words, the shape of the tamed process follows CHASE dynamical system.
stochastic approximation
The approach is, of course, close to many classical treatment of the stochastically
perturbed dynamical systems, see e.g.[7]. It should be noted, that if one rescales the
polygons P(k) by t (to keep their linear sizes approximately constant), then the step
sizes become c/t, which is the typical scale of the algorithms of stochastic approximation (compare [13]).
4 Properties of CHASE Flow
We outline several properties of the CHASE dynamics, before analyzing their equilibria.
4.1 Symmetries
One of the attractive features of the CHASE dynamics is its high symmetries. Thus,
the Lemma 2 implies
Proposition 1 The CHASE dynamics defines a field of directions (i.e. a vector field
defined up to point-dependent rescaling) on the shape space VPk .
(We will be retaining the name for the field of direction on VPk obtained by the projection of CHASE .)
This means that we can track the shapes of the polygonal chains VPk . In particular, if there existed an exponentially stable critical point of the reduction of the
Convex Hull Asymptotic Shape Evolution
11
CHASE dynamics to the shape space, the tamed process would have a positive probability of converging to the corresponding shape.
However, as we will show below, there are no stable equilibria in the space VPk
for k ≥ 4.
CHASE as
the gradient for perimeter
Another interesting property of CHASE is the fact that it is a gradient flow, with respect to the natural Euclidean metric on VPk .
Proposition 2 The CHASE dynamics is the gradient of the function ψ(P) taking a
polygonal chain to the perimeter of its convex hull.
The proof is immediate: the variation of the length of the side [xil , xil+1 ] when xil
changes infinitesimally is the scalar product with
xil − xii+1
,
|xil − xii+1 |
whence the claim follows.
We remark here that if one chooses the other bisector
x˙il =
xi − xii−1
xil − xii+1
− l
,
|xil − xii+1 | |xil − xii−1 |
(2)
as the direction of the velocity, the resulting dynamics would preserve the perimeter
and is useful in building customized Birkhoff’s billiard tables, see [2].
4.2 Equilibria
According to Proposition 1, CHASE defines a vector field on VPk , and one would like
to understand which shapes are preserved by this dynamics. Quite obviously, the
regular k-polygons are preserved. Three more classes of shapes are preserved as
well, in fact, shown below: trapezoids, rhombi and drops obtained from a regular
2k-polygon by extending a pair of next-to-opposite sides.
As it turned out, these classes exhaust all possible shapes invariant under the
CHASE dynamics.
Theorem 2 The equilibrium points of CHASE in conf consist of the polygonal
chains whole convex hulls (having l ≤ k vertices) are regular l-sided polygons.
Proof. Via straightforward trigonometry (using the notation shown on the Figure 6)
we derive the rate of change of the angle φ j as given by
ϕ̇ j =
sin ϕ j+1 − sin ϕ j sin ϕ j−1 − sin ϕ j
+
,
`j
` j−1
(3)
12
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
Fig. 5. Left to right, shapes preserved by the CHASE dynamics: regular polygons, trapezoids,
rhombi, and drops. In pictures of rhombus and trapezoid, the dashed lines are the bisectors.
xj
lj
ϕ j−1
ϕj
l j+1
ϕj+1
Fig. 6. Notations for Theorem 2.
and, similarly, the rates for the lengths ` j :
`˙j = 2 + cos ϕ j + cos ϕ j−1 .
(4)
If the shape of the polygon remains invariant, the angles are constant. This implies a system of linear equations on s j := sin ϕ j ’s. It is immediate that this system is
identical to the conditions on stationary probabilities on a continuous time Markov
chain on the circular graph with transition rates 1/` j . As this auxiliary Markov chain
is manifestly ergodic, such stationary probabilities are unique and therefore the solutions are spanned by the vectors of (s j ) j = (1, 1, . . . , 1). Hence, all sin ϕ j are equal,
and that therefore for some ϕ, all of ϕ j are equal to either ϕ or π − ϕ.
We note that this condition implies that one of the following holds:
•
•
•
•
There are 3 acute angles, forcing the polygon to be a regular triangle;
or there are 2 acute angles equal to ϕ < π/2, and two complementary angles
π − ϕ, forcing the polygon to be either a trapezoid or a rhombus;
or there is one acute angle ϕ, and all the remaining angles are equal to π − ϕ, or
all angles are obtuse.
Convex Hull Asymptotic Shape Evolution
13
The 4-sided polygons are easily seen to be realizable with the angle ϕ being a continuous shape parameter.
As for the last two options, the preservation of the shape implies that the ratios of
˙ .
the lengths ` j /` j+1 remains constant, and therefore equal to the ratios of the `˙j /` j+1
Combining this with the equation (4) implies that the ratios of the lengths `i /` j can
take only following values (in case of one acute angle)
1,
1 ∓ cos ϕ
, (1 ± cos ϕ)±1 .
1 ± cos ϕ
This immediately lead to the conclusion that only drop shape can satisfy this condition.
As for the case where all the angles are equal, the equality of all sidelengths is
obvious.
4.3 Regular polygons are unstable
As we discussed in section 2.5, the space of shapes of k-sided polygonal chains
is naturally isomorphic to the k − 2-dimensional complex projective space and has
therefore (real) dimension 2k − 4.
A natural coordinate frame on a (everywhere dense) chart in this space is given
by the lengths of the sides of the k-gon, normalized to ∑ ` j = 1 (this gives k − 1
coordinates) and collection of angles ϕ j satisfying the condition ∑ ϕ j = (k − 2)π.
The condition of closing the polygon implies that only (k − 3) of these angles are
independent.
Under the CHASE dynamics the vertices of the k-sided polygon move according
to the equation (2). The next proposition shows that the regular k-sided polygon,
while a fixed point of CHASE dynamics, is unstable in linear approximation.
In the next proposition we talk about the linearization of the CHASE dynamics on
the shape space.
Remark that the vector field CHASE there is defined only up to a multiplication
by a positive smooth function. However, one can readily verify that the linearization
of the vector field at an equilibrium point is unaffected by this ambiguity.
Theorem 3 The linearization of the CHASE dynamics near the k-regular polygonal
chain the 2k − 4-dimensional space of k-sided shape has exactly (k − 3)-dimensional
stable subspace (tangent to the manifold {ϕ = const} and (k − 1)-dimensional invariant subspace, unstable for k ≥ 5.
Proof. One can check immediately that under the CHASE evolution, the sides of the
k-gons with all angles equal (to π(1 − 2/k)) move parallel to themselves. This proves
the invariance of {ϕ = const}, and also the fact that (when considered in the space of
k-sided polygons), the lengths are all growing linearly, with the same speed. Therefore, their ratios asymptotically tend to 1, proving the first claim.
It follows that in the (`, ϕ) frame on conf, the linearization of the CHASE vector
field is block-upper-triangular:
14
Maxim Arnold, Yuliy Baryshnikov and Steven M. LaValle
J=
.
J` . .
0 Jϕ
!
Using (3) we compute Jϕ , to obtain a cyclic tri-diagonal matrix:


2 −1 0 · · · 0 −1
 −1 2 −1 0 · · · 0 


(k−2)π  0 −1 2 −1 · · ·
0
2 cos( k ) 

Jϕ = −
 ..
..  ,
.. .. ..
 .

`
.
.
.
.


 0 · · · 0 −1 2 −1 
−1 0 · · · 0 −1 2
whence
spec(Jϕ ) j = −
4 cos( (k−2)π
)
2π j
k
1 − cos
`
k
which belongs to (0, ∞) if k ≥ 5. This, clearly, proves the claim.
Stability of triangles
A corollary of the proof of Theorem 3 implies that the regular triangles, under the
CHASE dynamics are stable. This result is, however, relatively useless, as the Markov
chain Pt is rather different from the tamed dynamics (which is approximated by
CHASE ): the triangles acquire extra vertices, and, while the growth of the resulting
polygons might or might not be slowed down compared to the tamed process, we are
missing currently tools to analyze it.
For this reason we present a simple analysis of somewhat more general than
CHASE dynamics, where the averaged speed of an apex at j-th point ( j = 1, 2, 3) has
a outward velocity v = v(ϕi /2) directed along the bisector, but having a general form
(not necessarily 2 cos ϕ/2, as in the case of CHASE ).
We will write
α = ϕ1 /2, β = ϕ2 /2, γ = ϕ3 /2.
As above, we find
α̇ =
1
1
(v(γ) sin γ − v(α) sin α) +
(v(β ) sin β − v(α) sin α)
sin β
sin γ
(5)
Denote g(α) = v(α) sin α. Then, condition on stationary point gets the form
g(α) = g(β ) = g(γ).
Differentiating (5) with respect to α and β we obtain linear stability condition
on stationary point. Let J = (Ji, j ) be the linear part of (5).
Then the linear stability conditions have the form
1
1
1
1
1
1
0
0
0
+
+ g (β )
+
+ g (γ)
+
0 > tr(J) = − g (α)
sin β sin γ
sin γ sin α
sin α sin β
Convex Hull Asymptotic Shape Evolution
15
and
0 6 det(J) = g0 (α)g0 (β ) + g0 (β )g0 (γ) + g0 (γ)g0 (α)
1
1
1
+
+
sin α sin β sin β sin γ sin γ sin α
In particular, if g0 (π/3) > 0, the regular triangles are stable.
5 Conclusions
5.1 Summary
Summarizing, the results we presented lend theoretical support to the observed experimentally phenomena: the CHASE dynamics which is approximating the original
Markov chain well for large polygons near k-sided regular ones for k ≥ 5 is unstable.
While we do not know what is the correct approximation for the polygons close (in
some sense) to large triangles, a reasonable CHASE -like approximation is stable near
regular triangles.
Clearly, the problem of proving the asymptotic symmetry breaking is still open.
5.2 Higher dimensions
One can look at the analogous problem in higher-dimensional setting. Experiments
show that the convex hulls of RRTs in Rd form approximately a regular d-simplex.
We do not have, yet, any results similar to our planar case.
5.3 Case k → ∞
Let us now consider limiting case for the infinitely many vertices of initial shape.
Without going into details, we just remark here, that in a natural parametrization,
the CHASE evolution is described in this limit by the one-dimensional Boussinesq
equation
ϕ̇ = (ϕ 2 )00 .
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